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Tomoyuki ICHIBA

Associate Professor, Department of Statistics & Applied Probability and Center for Financial Mathematics and Actuarial Research, and Mathematics in College of Creative Studies

at University of California Santa Barbara

Current Research

This project studies some topics in the system of Stochastic Differential Equations with possibly discontinuous and degenerate diffusion coefficients with applications to stochastic portfolio management and optimization problems. Here the coefficients of the equations are functions of the states and determine the stochastic behavior of the system. The coefficients, the initial states and the stochastic noise of the system are main inputs of the model. The output of the system is the stochastic process which can be construed as the dynamics of the financial system. The long-term goals of the proposed activity are two-folds: (i) to defend against financial crises and (ii) to provide efficient investment strategies in the financial markets.

Preprints

(1) ``Large deviations for interacting Bessel-like processes and applications to systemic risk" (with M. Shkolnikov) arXiv1303.3061

(2) ``Stochastic integral equations for Walsh semimartingales" (with Ioannis Karatzas, Vilmos Prokaj and Minghan Yan) arXiv: 1505.02504

(3) ``Yet another condition for absence of collisions for competing Brownian particles" (with Andrey Sarantsev) arXiv: 1608.07220

Recently Published Papers

[11] ``Skew-unfolding the Skorokhod reflection of a continuous semimartingale" Stochastic Analysis and Applications 2014 In Honor of Terry Lyons. Springer Proceedings in Mathematics & Statistics (2014) 349-376 (with Ioannis Karatzas) arXiv: 1404.4662.

[10] ``Planar diffusions with rank-based characteristics and perturbed Tanaka equations" Probability Theory and Related Fields 156 (2013) 343-374; a full version with additional topics of Transition probabilities, Time reversibility and Maximality is also available in (with E. R. Fernholz, V. Prokaj and I. Karatzas) arXiv:1108.3992 .

[9] ``Strong solutions of stochastic equations with rank-based coefficients" Probability Theory and Related Fields 156 (2013) 229-248 (with I. Karatzas and M. Shkolnikov) arXiv1109.3823 .

[8] ``Diffusions with rank-based characteristics and values in the nonnegative quadrant" Bernoulli 19 (2013) 2455-2493 (with Ioannis Karatzas and Vilmos Prokaj) arXiv:1202.0036.

[7] ``Convergence rates for rank-based models with applications to portfolio theory" Probability Theory and Related Fields 156 (2013) 415-448 (with S. Pal and M. Shkolnikov) arXiv:1108.0384.

[6] ``A second-order stock market model" Annals of Finance 9 (2013), 439-454 (with R. Fernholz and I. Karatzas) arXiv:1302.3870.

[5] ``Two Brownian particles with rank-based characteristics and skew-elastic collisions'' Stochastic Processes and their Applications 123 (2013) 2999-3026 (with E. Robert Fernholz and Ioannis Karatzas) arXiv:1206.4350 .

[4] ``Stability in a model of inter-bank lending" SIAM Journal of Financial Mathematics 4 (2013) 784-803 (with J-P. Fouque).

[3] ``Hybrid Atlas models" Annals of Applied Probability 21 no 2. (2011), 609-644. (with R. Fernholz, I. Karatzas, A. Banner and V. Papathanakos) arXiv:0909.0065.

[2] ``Collisions of Brownian particles" Annals of Applied Probability 20 no 3. (2010), 951-977 (with I. Karatzas) arXiv:0810.2149.

[1] ``Efficient estimation of one-dimensional diffusion first passage time densities via Monte Carlo simulation" Journal of Applied Probability 48 no 3. (2011) 699-712 (with Constantinos Kardaraz) arXiv:1008.1326.

Dissertation

``Topics in multi-dimensional diffusions: attainability, reflection, ergodicity and rankings'' Dissertation Columbia University. 2009. [924kb].

Conference Proceedings

[C2] `` Folding and Skew-Unfolding of One-dimensional Continuous Semimartingales" RIMS Kokyuroku No 1952: Symposium on Probability Theory, Kyoto 2014

[C1] ``On planar rank-based diffusions with skew-elastic collisions" RIMS Kokyuroku No 1903: Symposium on Probability Theory, Kyoto 2013

Posters

``Stochastic Integral Equations for Walsh Semimartingales" IMA Special Workshop: Reflected Brownian motions, Stochastic Networks and their applications at University of Minnesota (June 2015)

``Strong/Weak Solutions of 2D Diffusions with Rank-based Characterisits" Probabiliy, Control and Finance at Columbia University (May 2012)

Miscellaneous Papers of Statistics in Sports, Biology and Actuarial Science

(S4) ``Estimating the effect of the red card in soccer: when to commit an offense in exchange for preventing a goal opportunity'' Journal of Quantitative Analysis in Sports 5, no. 1 / 2009. (with Jan Vecer and Frantisek Kopriva).

(S3)``On probabilistic excitement of sports games.'' Journal of Quantitative Analysis in Sports 3, no. 6 / 2007. (with Jan Vecer and Mladen Laudanovic).

(S2) ``Assessing substitution variation across sites in grass chloroplast DNA.'' Journal of Molecular Evolution 64, no. 6 pp. 605-613 / June, 2007. (with Tian Zheng and Brian Morton) .

(S1) ``On multi-period statistical risk management methods and equity-linked life insurance.'' Journal of the Japan Statistical Society. Japanese issue 35, no. 2 pp. 103-123. / 2006. abstract. (with Naoto Kunitomo).

Last modified: August 2015.

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